Q.4 - The Practice of Statistics for AP

Question

 Flu vaccine A drug company has developed a new vaccine for preventing the flu. The company claims that fewer than $5\%$ of adults who use its vaccine will get the flu. To test the claim, researchers give the vaccine to a random sample of $1000$ adults. Of these, $43$ get the flu.

(a) Do these data provide convincing evidence to support the company's claim? Perform an appropriate test to support your answer.

(b) Which kind of mistake - a Type I error or a Type II error-could you have made in (a)? Explain.

(c) From the company's point of view, would a Type I error or Type Il error be more serious? Why?

Step-by-Step Solution

Answer

(a) No

(b) Type ll error

(c)Type l error

Part (a) Step1 :Given Information

Given in the question that,

Number of trials$\left(n\right)=1000$

Number of success$\left(x\right)=43$

Population proportion$\left(p\right)=0.05$ we have to ensure  these data provide convincing evidence to support the company's claim.

Part (a) Step2: Explanation

The formula to complete the test statistic is:

$z=\frac{\hat{p}-p_0}{\sqrt{{\displaystyle \frac{p_0\left(1-p_0\right)}{n}}}}$

using the information the null and alternative hypothesis are:

$H_0:P=0.05$

$H_O:P_0<0.05$

Using the Ti-83 calculator the output is:

The p-value is higher than the level of significance. As a result, the non-rejection of the null hypothesis is represented as decision.

There is adequate evidence to support the stated assertion at the $5\%$significance level.

Part(b) Step 1: Given Information

Given in the question that Flu vaccine A drug company has developed a new vaccine for preventing the flu. The company claims that fewer than $5\%$ of adults who use its vaccine will get the flu. To test the claim, researchers give the vaccine to a random sample of $1000$ adults. Of these, $43$ get the flu. we have to find Which kind of mistake - a Type I error or a Type II error-could you have made in (a).

Part (b) Step 2: Explanation

From above part, the null hypothesis has not been rejected. Thus, there is a possibility of committing the Type ll error.

Part (c) Step 1: Given Information

Given in the question that Flu vaccine A drug company has developed a new vaccine for preventing the flu. The company claims that fewer than $5\%$ of adults who use its vaccine will get the flu. To test the claim, researchers give the vaccine to a random sample of $1000$ adults. Of these, $43$ get the flu. we have to find From the company's point of view, would a Type I error or Type Il error be more serious .

Part (c) Step2: Explanation

It's likely that more people are acquiring flu as a result of the Type I error than the test indicated. As a result, type I errors are more serious.